few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Visualize relationships between two-dimensional and three-dimensional objects.

Understand and apply theorems about circles.

Apply trigonometry to general triangles.

Understand similarity in terms of similarity transformations.

Find arc lengths and areas of sectors of circles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Reason with shapes and their attributes.

Explain volume formulas and use them to solve problems.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Solve real-world and mathematical problems involving area, surface area, and volume.

Prove theorems involving similarity.

Translate between the geometric description and the equation for a conic section.

Understand congruence in terms of rigid motions.

Experiment with transformations in the plane.

Classify two-dimensional figures into categories based on their properties.

Analyze, compare, create, and compose shapes.

Apply geometric concepts in modeling situations.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Understand and apply the Pythagorean Theorem.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Use coordinates to prove simple geometric theorems algebraically.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Prove geometric theorems.

Make geometric constructions.

Define trigonometric ratios and solve problems involving right triangles.
problem_activity: Task
This problem examines equations defining different circles in the $x$-$y$ plane.
Use the Pythagorean theorem to
find an equation in $x$ and $y$ whose solutions are the points on the circle  of radius 2 with center $(1,1)$  and explain why it works.
Suppose $r$ is a positive number and $(a,b)$ a point in the plane.
Use the Pythagorean theorem to
find an equation in $x$ and $y$ whose solutions are the points on the circle of radius $r$ with center $(a,b)$ and explain why it works.


correct_option_index: 13